The provider does not have to take down the content. It's not doing so only affects whether it can raise the safe harbor defense of the DMCA in a subsequent suit. A simple reading of the notice would have revealed that it was invalid and thus could not possibly sustain a suit for copyright infringement. Indeed, since the DMCA safe harbor doesn't protect a hosting company from being sued for trademark infringement, the hosting company didn't even really cover its ass, legally, in this situation. If GoPro had sued the hosting provider for trademark infringement, its responding to the DMCA takedown notice would've had no relevance.